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 A301830 Number of factorizations of n into factors (greater than 1) of two kinds. 19
 1, 2, 2, 5, 2, 6, 2, 10, 5, 6, 2, 16, 2, 6, 6, 20, 2, 16, 2, 16, 6, 6, 2, 36, 5, 6, 10, 16, 2, 22, 2, 36, 6, 6, 6, 46, 2, 6, 6, 36, 2, 22, 2, 16, 16, 6, 2, 76, 5, 16, 6, 16, 2, 36, 6, 36, 6, 6, 2, 64, 2, 6, 16, 65, 6, 22, 2, 16, 6, 22, 2, 108, 2, 6, 16, 16, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..10000 Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020. FORMULA Dirichlet g.f.: Product_{n > 1} 1/(1 - n^(-s))^2. [corrected by Ilya Gutkovskiy, Dec 14 2020] a(p^n) = A000712(n) for prime p. - Andrew Howroyd, Nov 18 2018 EXAMPLE The a(6) = 6 factorizations: (2*3)*(), (3)*(2), (2)*(3), ()*(2*3), (6)*(), ()*(6). The a(12) = 16 factorizations:   ()*(2*2*3), (2)*(2*3), (3)*(2*2), (2*2)*(3), (2*3)*(2), (2*2*3)*(),   ()*(2*6), (2)*(6), (6)*(2), (2*6)*(), ()*(3*4), (3)*(4), (4)*(3), (3*4)*(),   ()*(12), (12)*(). MATHEMATICA facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]]; Table[Sum[Length[facs[d]]*Length[facs[n/d]], {d, Divisors[n]}], {n, 100}] PROG (PARI) MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v} seq(n)={MultEulerT(vector(n, i, 2))} \\ Andrew Howroyd, Nov 18 2018 CROSSREFS Cf. A000712, A001055, A001222, A001405, A122768, A276024, A281113, A284640, A295632, A299701, A299702, A299729, A301829. Sequence in context: A240081 A305791 A299764 * A305799 A294339 A185291 Adjacent sequences:  A301827 A301828 A301829 * A301831 A301832 A301833 KEYWORD nonn AUTHOR Gus Wiseman, Mar 27 2018 STATUS approved

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Last modified January 27 16:41 EST 2022. Contains 350611 sequences. (Running on oeis4.)